Two types of materials can exhibit negative refraction and thus possess negative refractive indices. One is the wire and split-ring resonator metamaterial. The other is the photonic crystal.
A metamaterial has a crystal structure which has a lattice spacing much less than the operating wave length. The mechanism is to invoke electric and magnetic resonances so that this artificial structure can have negative permittivity and permeability simultaneously. Since the material has negative refraction only in the neighborhood of resonances, high loss is expected. This severely restricts the application of this type of material.
A photonic crystal uses a completely different mechanism to achieve negative refraction. A photonic crystal is a crystal with periodic variation of either permittivity such as dielectric photonic crystals or permeability such as magnetic photonic crystals. With high contrast of the dielectric constants of the component materials of the photonic crystal, the frequency and wave vector dispersion relation is strongly modulated compared with homogeneous media. In general the wave vector and the group velocity in the crystal are neither parallel nor anti-parallel to each other. At certain frequencies, the group velocity and/or wave vector refract negatively. A photonic crystal has the advantage that the design can be scaled to any frequency from radio frequencies to optical, and even X-rays.